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Monge points, Euler lines, andFeuerbach spheres in Minkowski spaces
Undine Leopold and Horst Martini
February 22, 2016
Dedicated to Egon Schulte and Karoly Bezdek on the occasion of their60th birthdays
Abstract: It is surprising, but an established fact that the field of Ele-mentary Geometry referring to normed spaces (= Minkowski spaces) is nota systematically developed discipline. There are many natural notions andproblems of elementary and classical geometry that were never investigatedin this more general framework, although their Euclidean subcases are wellknown and this extended viewpoint is promising. An example is the geom-etry of simplices in non-Euclidean normed spaces; not many papers in thisdirection exist. Inspired by this lack of natural results on Minkowskiansimplices, we present a collection of new results as non-Euclidean gen-eralizations of well-known fundamental properties of Euclidean simplices.These results refer to Minkowskian analogues of notions like Euler line, or-thocentricity, Monge point, and Feuerbach sphere of a simplex in a normedspace. In addition, we derive some related results on polygons (instead oftriangles) in normed planes.Keywords and phrases: Birkhoff orthogonality, centroid, circumsphere,Euler line, Feuerbach sphere, isosceles orthogonality, Mannheim’s theorem,Minkowskian simplex, Monge point, normality, normed space, orthocen-tricity
2010 Mathematics Subject Classification: 46B20, 51M05, 51M20,52A10, 52A20, 52A21, 52B11
1 Introduction
Looking at basic literature on the geometry of finite dimensional real Banach spaces(see, e.g., the monograph [49] and the surveys [39] and [31]), the reader will observe thatthere is no systematic representation of results in the spirit of elementary and classicalgeometry in such spaces (in other words, the field of Elementary Geometry is not really
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developed in normed spaces, also called Minkowski spaces). This is not only meant inthe sense that a classifying, hierarchical structure of theorems is missing. Also, it isalready appealing to find the way of correctly defining analogous notions. An exampleof such a non-developed partial field is the geometry of simplices in non-EuclideanMinkowski spaces. Inspired by this indicated lack of natural results on Minkowskiansimplices, we derive a collection of new results which reflect non-Euclidean analoguesand extensions of well known properties of Euclidean simplices. These results arebased on, or refer to, generalizations of notions like Euler lines, orthocentricity (ofcourse depending on a suitable orthogonality notion), Monge points, and Feuerbachspheres of simplices in Minkowski spaces. It should be noticed that some of thesetopics are even not established for Minkowski planes; most of our results are derivedimmediately for simplices in Minkowski spaces of arbitrary finite dimension.
In plane Euclidean geometry, the Euler line of a given triangle is a well-studied objectwhich contains many interesting points besides the circumcenter and the centroid ofthis triangle. Other special points on the Euler line include the orthocenter and thecenter of the so-called nine-point- or Feuerbach circle. Notions like this can be extendedto simplices in higher dimensional Euclidean space, and the respective results cansometimes be sharpened for important subfamilies of general simplices, like for examplethe family of orthocentric simplices. Using new methods developed by Grassmann forstudying the d-dimensional Euclidean space, this was done already in the 19th century.Two early related references are [42] and [44]. Deeper results were obtained later; theconcept of Euler line and some related notions have been generalized to Euclideanhigher dimensional space in [9,15–18,22,23,25,27,43] for orthocentric simplices, and in[10,14,19,30,43,46] for general simplices. Other interesting generalizations in Euclideangeometry refer to Euler lines of cyclic polygons, see [24]. For a few results in Minkowskiplanes and spaces we refer to [6, 8, 11, 34, 40]. The Feuerbach circle of a triangle inthe Euclidean plane passes through the feet of the three altitudes, the midpoints ofthe three sides, and the midpoints of the segments from the three vertices to theorthocenter of that triangle. Beautiful generalizations of the Feuerbach circle to d-dimensional Euclidean space for orthocentric simplices have been obtained in [9,18,23,25], and for general simplices in [10, 19, 43]. Minkowskian analogues have so far onlybeen discussed in normed planes, see [8, 34,45].
A d-dimensional (normed or) Minkowski space (Rd, ‖ · ‖) is the vector space Rd
equipped with a norm ‖ · ‖. A norm can be given implicitly by its unit ball B(O, 1),which is a convex body centered at the origin O; its boundary S(O, 1) is the unitsphere of the normed space. Any homothet of the unit ball is called a Minkowskianball and denoted by B(X, r), where X is its center and r > 0 its radius; its boundaryis the Minkowskian sphere S(X, r). Two-dimensional Minkowski spaces are Minkowskiplanes, and for an overview on what has been done in the geometry of normed planesand spaces we refer to the book [49], and to the surveys [39] and [31].
The fundamental difference between non-Euclidean Minkowski spaces and the Eu-clidean space is the absence of an inner product, and thus the notions of angles andorthogonality do not exist in the usual sense. Nevertheless, several types of orthogo-nality can be defined (see [1], [2], and [5] for an overview), with isosceles and Birkhofforthogonalities being the most prominent examples. We say that y is isosceles or-
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Figure 1: A triangle with several circumcenters (left), and a triangle without a circum-center (right), as illustrated by suitable homothets of the unit ball.
thogonal to x, denoted x ⊥I y, when ‖x + y‖ = ‖x − y‖. Isosceles orthogonalityis thus the orthogonality of diagonals in a parallelogram with equal side lengths (arhombus in Euclidean space). It is also the orthogonality of chords over a diameter.By contrast, y is Birkhoff orthogonal to x, denoted x ⊥B y, when ‖x‖ ≤ ‖x + αy‖for any α ∈ R. Thus Birkhoff orthogonality is the (unsymmetric) orthogonality of aradius x and corresponding tangent vector y of some ball centered at the origin O.For hyperplanes and lines, there is the notion of normality. A direction (vector) v isnormal to a hyperplane E if there exists a radius r > 0, such that E supports the ballB(O, r) at a multiple of v. Equivalently, v is normal to E if any vector parallel to Eis Birkhoff orthogonal to v.
For any two distinct points P , Q, we denote by [PQ] the closed segment, by 〈PQ〉the spanned line (affine hull), and by [PQ〉 the ray {P + λ(Q− P ) | λ ≥ 0}; we write‖[PQ]‖ for the length of [PQ]. We will use the usual abbreviations aff, conv, ∂, andcone for the affine hull, convex hull, boundary and cone over a set, respectively.
In this article, we focus on the geometry of simplices in d-dimensional Minkowskispaces. As usual, a d-simplex is the convex hull of d + 1 points in general linearposition, or the non-empty intersection of d+ 1 closed half-spaces in general position.We underline that by circumcenters of simplices we mean the centers of circumspheres(or -balls) of simplices, i.e., of Minkowskian spheres containing all the vertices of therespective simplex (see, e.g., [3]). A related, but different notion is that of minimalenclosing spheres of simplices, sometimes also called circumspheres (cf., e.g., [4]); thisnotion is not discussed here. In the two-dimensional situation, circumspheres and -balls are called circumcircles and -discs. In Minkowski spaces, simplices may haveseveral, precisely one, or no circumcenter at all, depending on the shape of the unitball, see Figure 1. Examples without circumcenters may only be constructed for non-smooth norms, as all smooth norms allow inscription into a ball [20,28]. We focus onthe case where there is at least one circumcenter.
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2 Orthocentric simplices and the Monge point inEuclidean space
We begin with a short survey on results related to orthocentricity in Euclidean space.In Euclidean geometry, not every simplex in dimension d ≥ 3 possesses an orthocenter,i.e., a point common to all the altitudes. However, if such a point H exists, the simplexis called orthocentric and possesses a number of special properties (compare the surveycontained in [15] and [22]). The following proposition is well known (see again [15]).
Proposition 2.1. A d-simplex T in Euclidean space is orthocentric if and only if thedirection of every edge is perpendicular to the affine hull of the vertices not in thatedge (i.e., the affine hull of the opposite (d − 2)-face). Equivalently, a d-simplex inEuclidean space is orthocentric if and only if any two disjoint edges are perpendicular.
The (d−2)-faces of a d-polytope are sometimes called ridges, see [41]. The followingfact (see also the survey in [15]) can be proved in many ways, and has been posed as aproblem in the American Mathematical Monthly [26]. Note that orthocenters are notdefined for an edge or a point.
Proposition 2.2. In an orthocentric Euclidean d-simplex (d ≥ 3), the foot of everyaltitude is the orthocenter of the opposite facet.
In absence of a guaranteed orthocenter, the literature on Euclidean geometry (e.g.[7, 13] for three dimensions, [10, 15, 22] for the general case) defines the Monge pointof a tetrahedron or higher-dimensional simplex as the intersection of so-called Monge(hyper-)planes. The Monge point coincides with the Euclidean orthocenter if the latterexists [7, 10, 13]. From this, theorems about the Euler line, the Feuerbach circle, etc.can be generalized to higher dimensional simplices, see all the references given in theIntroduction, and see Section 4 for Minkowskian analogues. We recall the definitionand the following theorems from [10].
Definition 2.1. Let T be a d-simplex in Euclidean d-space. A Monge hyperplane is ahyperplane which is perpendicular to an edge of the simplex and which passes throughthe centroid of the opposite (d− 2)-face (ridge).
Theorem 2.1. (Monge Theorem) The Monge hyperplanes of a Euclidean d-simplexhave precisely one point in common, which is called the Monge point N of the simplex.
Theorem 2.2. (Orthocenter Theorem) In an orthocentric Euclidean d-simplex, theMonge point N coincides with the orthocenter H.
Theorem 2.3. (Mannheim Theorem, see [7, 13] for d = 3, and [10] for arbitrary d)For any d-simplex, the d + 1 planes, each determined by an altitude of a d-simplexand the Monge point of the corresponding facet, pass through the Monge point of thed-simplex.
Regular simplices are orthocentric. Regular simplices are also equilateral, i.e., alltheir edges have equal length, as well as equifacetal, which means that all their facets
4
are isometric (congruent). Furthermore, the circumcenter M , centroid G, orthocenterH, and incenter I, i.e., the center of the unique inscribed sphere touching all facets,coincide. Conversely, we have the following statement, see [15].
Theorem 2.4. A Euclidean d-simplex T is regular, if and only if any of the followingconditions are fulfilled:
1. T is equilateral.
2. T is orthocentric and any two of the centers M , G, I, H coincide.
3. T is orthocentric and equifacetal.
As we will see in the next Section, the concept of Monge point generalizes to arbitraryMinkowski spaces, at least for simplices with a circumcenter.
3 The Monge point of simplices in Minkowski spaces
In this section, we generalize the definition of Monge point and Monge hyperplanes toMinkowski spaces of arbitrary (finite) dimension d ≥ 2.
Definition 3.1. Let (Rd, ‖ · ‖) be a d-dimensional Minkowski space, and let T be ad-simplex with a circumcenter M . For each pair (F,EF ) of a ridge F and oppositeedge EF , and if M is not the midpoint of EF , define the associated Monge line as theline through the centroid of F which is parallel to the line through M and the midpointof EF .
Theorem 3.1. Let (Rd, ‖ · ‖) be a d-dimensional Minkowski space, and let T be ad-simplex with a circumcenter M . Then the Monge lines of T are concurrent in asingle point NM , called the Monge point of T .
Before proving the theorem, essentially following the outline for Euclidean spacein [10], we first define a quasi-median of a d-simplex as a line joining the centroid ofa (d − 2)-face of the simplex with the midpoint of the opposite edge. The followingLemma was proved in [10] for Euclidean space, yet due to the definition of the centroidit holds true in any Minkowski space.
Lemma 3.1. The quasi-medians of a d-simplex T intersect in its centroid. The cen-troid divides each quasi-median in the ratio 2: (d − 1) (with the segment measuringd−1d+1 of the length of the quasimedian ending in the midpoint of an edge).
Proof. (Proof of the theorem) The proof is similar to, but more general than, the onein [10] for Euclidean space. First, for each (d − 2)-face F denote its centroid G(F ),and let G(EF ) be the midpoint or centroid of the opposite edge EF . Since a d-simplexpossesses
(d+12
)edges (ridges) and M can be located at the midpoint of at most one
of them, the auxiliary lines 〈MG(EF )〉 are well-defined for at least(d+12
)− 1 pairs
(F,EF ). The auxiliary line 〈MG(EF )〉, if well-defined, is parallel to the associatedMonge line 〈G(F )L(F )〉 of (F,EF ), where we define L(F ) := G(F ) + G(EF ) −M .
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MG(EF )
G
G(F )L(F )N(F )
Figure 2: Location of the Monge point.
Second, if M = G, then G and G(F ) both lie on 〈MG(EF )〉, i.e., each auxiliaryline coincides with the associated Monge line, and all these lines intersect in NM :=M = G (and this is the only point, since different edge midpoints define different lines〈MG(EF )〉). If M 6= G, then auxiliary line and Monge line are distinct. Observethat each quasimedian [G(F )G(EF )] connects a Monge line and the correspondingauxiliary line. The centroid G of the simplex T divides each quasimedian in the ratio2: (d−1), so the same division ratio holds true for the segment [MN(F )] which passesthrough the given circumcenter M , the centroid G of T , and ends at the point N(F )on [G(F )L(F )〉, see Figure 2. As a consequence of this common ratio, all points N(F )are indeed the same point NM , solely dependent on the chosen circumcenter (and thegiven simplex), and all rays [G(F )L(F )〉 meet at NM .
In keeping with the tradition in Euclidean space, we want to reformulate the theoremin terms of hyperplanes.
Definition 3.2. Let (Rd, ‖ · ‖) be a d-dimensional Minkowski space, and let T be a d-simplex with a circumcenter M . Suppose M is not the midpoint of an edge EF oppositea (d − 2)-face F of the simplex. For the pair (F,EF ) define the auxiliary pencil ofhyperplanes through M and the midpoint of EF . Furthermore, define the associatedMonge hyperplane pencil for the pair (F,EF ) as the translate of the auxiliary pencilsuch that all hyperplanes go through the centroid of F .
Corollary 3.2. Let (Rd, ‖ · ‖) be a d-dimensional Minkowski space, and let T be ad-simplex with a circumcenter M . Then the hyperplanes of all (well-defined) Mongehyperplane pencils of T intersect in a single point, namely the Monge point of T .
The following corollary tells us the precise location of the Monge point with respectto the vertices of the simplex and the given circumcenter.
Corollary 3.3. Let T = conv{A0, . . . , Ad} be a d-simplex in d-dimensional Minkowskispace, possessing a circumcenter M . Then the associated Monge point is determinedas
NM = M +
∑di=0(Ai −M)
d− 1.
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Proof. Let F be a ridge of the simplex, opposite the edge EF , such that G(EF ) 6= M(i.e., the edge midpoint is distinct from M ; such an edge must exist). From the proofof Theorem 3.1 we deduce for M 6= G that
‖[MG(EF )]‖ : ‖[G(F )NM ]‖ = ‖[MG]‖ : ‖[GNM ]‖ = (d− 1) : 2.
Thus
NM = M + (d+ 1)G−Md− 1
= M +(d+ 1)
∑di=0(Ai−M)
d+1
d− 1= M +
∑di=0(Ai −M)
d− 1.
For M = G we obtain NM = M = G.
Remark 3.4. In Euclidean context, each Monge hyperplane passes through the cen-troid of a (d− 2)-face F and is perpendicular to the opposite edge EF (here oppositeedge means the edge between the two vertices not in the ridge F ). However, we seethat perpendicularity is not necessary, and any hyperplane containing the associatedMonge line as per our definition is suitable (provided the Monge line is well-defined).Therefore, while our Minkowskian Monge pencils contain the correct Monge hyper-planes in Euclidean context, we have the confirmation that orthogonality of lines andhyperplanes need not necessarily play a role when finding the Monge point. The con-cept of Monge point is even an affine concept, as the circumcenter property of M isused nowhere (i.e., any point M can be used to construct “Monge lines” intersectingat NM with the analytical expression given above).
In particular, we obtain the following corollary, which appears to be new also forthe Euclidean case.
Corollary 3.5. Let (Rd, ‖ · ‖) be a d-dimensional Minkowski space, and let T be ad-simplex with a circumcenter M . For each ridge F and the opposite edge EF withmidpoint G(EF ), if M 6= G(EF ) and 〈MG(EF )〉 is not parallel to F , define an M -hyperplane as the hyperplane containing F and being parallel to 〈MG(EF )〉. Then alldefined M -hyperplanes intersect in the Monge point NM .
Proof. Let A0, . . . , Ad denote the vertices of T . Observe that, since the medial hyper-planes of T are in general position, M lies in at most d of the d+1 medial hyperplanes.Without loss of generality, M does not lie in the medial hyperplane between A0 andits opposite facet. Since G([A0Ai]) lies in that medial hyperplane for i = 1, . . . , d, andthe ridge F0,i opposite [A0Ai] is parallel to that medial hyperplane, we conclude that[MG([A0Ai])] is not parallel to F0,i, and the M -hyperplanes are defined at least forthe d pairs (F0,i, [A0Ai]).
Consider the (d− 1)-simplex
T0 := conv{G([A0Ai]), i = 1, . . . , d},
which is a homothet of the facet F0 of T opposite A0 with homothety center A0
and factor 12 . The related (d − 1)-simplex T ′0 is obtained by homothety of T0 in
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G(T0) = A0+G(F0)2 and with homothety factor −(d − 1). Observe that the (d − 2)-
dimensional facets of T ′0 (ridges of T ) pass through the vertices of T0 and are parallelto the (d− 2)-dimensional facets of T0.
Now, the d M -hyperplanes previously considered are parallel to the hyperplanesdefined by the facets of the d-simplex
conv{M ∪ T ′0}
through the vertex M . Therefore, these M -hyperplanes are in general position, inter-secting only in the Monge point NM which, by definition, is contained in every definedM -hyperplane.
Another theorem concerning the Monge point in Euclidean space is the Mannheimtheorem, see [13] for the three-dimensional case and [10] for generalizations. It is ourTheorem 2.3 above, and it presents an example of a statement that cannot be extendedto Minkowski spaces. The simple reason is that hyperplane sections of Minkowskianballs need not be centrally symmetric. Therefore, in general the concept of Mongepoint of a d-simplex cannot be transferred to its facets.
4 Euler lines and generalized Feuerbach spheres ofMinkowskian simplices
We define as Euler line associated to a circumcenter M the straight line connectingM with the centroid G. Thus, in the case of the centroid being a circumcenter,the associated Euler line is not well-defined. We now consider the situation in d-dimensional Minkowski space for d ≥ 2.
Definition 4.1. For a d-simplex T := conv{A0, . . . , Ad} with circumcenter M , definethe complementary line of a facet with respect to M as the translate of the line betweenthe circumcenter M of the simplex and the centroid of the facet, passing through theopposite vertex. If A1, . . . , Ad are the vertices of the chosen facet with centroid G0,then the complementary line is A0 + t · (G0 −M), t ∈ R.
Remark 4.1. As in the planar case, for smooth norms such a circumcenter alwaysexists (see [29], and [39, §7.1]). For a non-smooth norm, simplices without a circum-center may exist (see again Figure 1 (right) for the planar situation, and it is easy toconstruct examples also for general d).
The following theorem is an easy consequence of the definition of the centroid.
Theorem 4.1. The complementary lines of the facets of a d-simplex T with respect toa fixed circumcenter M connect all the vertices to the same point, the complementarypoint PM associated to M .
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Proof. Let T = conv{A0, . . . , Ad}, and let Gj denote the centroid of the facet oppositevertex Aj . Then the point
PM = M+
d∑i=0
(Ai−M) = Aj+d
d∑i=0i 6=j
Ai
d−M
= Aj+d(Gj−M) for each j = 0, . . . d,
lies on each complementary line. For each j = 0, . . . , d, we have ‖[PMAj ]‖ = d‖MGj‖.
Various useful types of orthogonalities have been defined in Minkowski spaces forpairs of vectors, all coinciding with the usual orthogonality in Euclidean space, yet weonly have normality as a concept for vectors and (hyper-)planes. We call each segment[PMAj ] on a complementary line the complementary segment associated to the oppo-site facet. As such, a complementary segment is not orthogonal to a hyperplane in anyknown sense. However, in dimension two we obtain the familiar isosceles orthogonalitybetween an edge of a simplex (triangle side) and the corresponding complementary seg-ment (orthogonality if we are in the Euclidean plane!), and the complementary pointis the C-orthocenter [8, 34]. Unlike the C-orthocenter, the notion of complementarypoint generalizes to any higher dimension.
Remark 4.2. The complementary point is even an affine notion, as we only useddivision ratios of segments on a line. The point PM can be constructed for any point M(circumcenter or not) in the following way: take the line connecting M to the centroidof a simplex facet (if distinct from M), and then consider the translated line passingthrough the vertex opposite the chosen facet. All lines of the latter kind intersectin a point (denoted PM in the present article), which has already been observed bySnapper [46].
The complementary point and Monge point associated to a simplex with circum-center M possess the following properties.
Theorem 4.2. Let T be a d-simplex (d ≥ 2) in Minkowskian space (Rd, ‖ · ‖), with acircumcenter M distinct from its centroid G.(a) The associated complementary point PM and the Monge point NM lie on the Eulerline 〈MG〉.(b) The centroid G divides the segment [MPM ] internally in the ratio 1 : d.(c) The associated Monge point NM divides the segment [MPM ] internally in the ratio1 : (d− 2).(d) The centroid G divides the segment [MNM ] internally in the ratio (d− 1) : 2.
Proof. Let T = conv{A0, . . . , Ad}. That the Euler line 〈GM〉 associated to M passes
9
through NM and PM can be seen from the following equations:
G =
∑di=0Ai
d+ 1= M +
∑di=0(Ai −M)
d+ 1,
NM = M +
∑di=0(Ai −M)
d− 1,
PM = M +
d∑i=0
(Ai −M).
Thus (a) is proved. The above equations also immediately prove (b) and (c). Proving(d) is an easy exercise in arithmetic:
‖G−M‖ : ‖NM −G‖ =
∥∥∥∥∥∑d
i=0(Ai −M)
d+ 1
∥∥∥∥∥ :
∥∥∥∥∥∑d
i=0(Ai −M)
d− 1−∑d
i=0(Ai −M)
d+ 1
∥∥∥∥∥=
∥∥∥∥∥∑d
i=0(Ai −M)
d+ 1
∥∥∥∥∥ :
∥∥∥∥∥2∑d
i=0(Ai −M)
(d− 1)(d+ 1)
∥∥∥∥∥= (d− 1) : 2.
Remark 4.3. We see that NM can be obtained by homothety of M from center G,with homothety ratio − 2
d−1 . Moreover, recall the M -hyperplanes from Corollary 3.5which intersect in NM . The above homothety takes each M -hyperplane to a certainparallel hyperplane through M . It turns out that these central planes (through thecircumcenter M) encompass the supporting hyperplanes through M of the auxiliarysimplex conv{M ∪ T ′0} in the proof of Corollary 3.5.
Considering the points of interest in Theorem 4.2, one may ask whether the point
M +∑d
i=0(Ai−M)
d on the Euler line, dividing [MPM ] internally in the ratio 1 : (d− 1),holds any special meaning. It turns out that it is the center of a sphere analogousto the well-known Feuerbach circle of a triangle in the Euclidean plane. The exten-sion to higher dimensional normed spaces for the case M 6= G is as follows (for the“degenerate” case M = G we refer to Corollary 4.8).
Theorem 4.3. (The 2(d + 1)- or Feuerbach sphere of a d-simplex) In an arbitraryMinkowski d-space, let T = conv{A0, . . . , Ad} be a d-simplex with a circumcenterM and circumradius R, and let G( 6= M) be its centroid. The sphere with center
FM := M +∑
(Ai−M)d+1i=1
d on the Euler line and of radius r := Rd passes through the
following 2(d+ 1) points:(a) the centroids Gi, i = 0, . . . , d, of the facets Fi of T (Fi is opposite vertex Ai), and(b) the points LM
i dividing the segments connecting the Monge point NM to the verticesAi of T , i = 0, . . . , d, in the ratio 1 : (d− 1).Moreover, S(FM , r) is a homothet of the circumsphere S(M,R) with respect to thecentroid G and homothety ratio −1d , i.e., G divides the segment [FMM ] internally inthe ratio 1 : d, and FM divides the segment [NMM ] internally in the ratio 1 : (d− 1).
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Remark 4.4. In analogy to the Feuerbach circle in the plane centered at the nine-point-center, we call FM the 2(d+ 1)-center of the simplex with respect to the circum-center M , and S(FM ,
Rd ) its Feuerbach or 2(d+ 1)-sphere.
Proof. The centroid of a facet opposite vertex Aj is Gj =
d∑i=0i6=j
Ai
d . We have R =‖Aj −M‖ for any j = 0, . . . , d, and thus
‖Gj − FM‖ =
∥∥∥∥∥∥∑d
i=0i6=j
Ai
d−M −
∑di=0(Ai −M)
d
∥∥∥∥∥∥ =
∥∥∥∥M −Aj
d
∥∥∥∥ =R
d,
which proves that S(FM ,Rd ) passes through the points in (a).
The Monge point is NM = M +∑d
i=0(Ai−M)
d−1 , thus
LMj := M +
∑di=0(Ai −M)
d− 1+Aj −M −
∑di=0(Ai−M)
d−1d
= M +(d− 1)
∑di=0(Ai −M)
d(d− 1)− M −Aj
d
= M +
∑di=0(Ai −M)
d− M −Aj
d.
Therefore,
‖LMj − FM‖ =
∥∥∥∥∥M +
∑di=0(Ai −M)
d− M −Aj
d−M −
∑di=0(Ai −M)
d
∥∥∥∥∥=
∥∥∥∥−M −Aj
d
∥∥∥∥ =R
d,
which proves that S(FM ,Rd ) passes through the points in (b). We also have
‖FM −G‖ : ‖G−M‖ =
∥∥∥M +∑d
i=0(Ai−M)
d −M −∑d
i=0(Ai−M)
d+1
∥∥∥∥∥∥M +∑d
i=0(Ai−M)
d+1 −M∥∥∥
=
∥∥∥∥∥∑d
i=0(Ai −M)
d(d+ 1)
∥∥∥∥∥ :
∥∥∥∥∥∑d
i=0(Ai −M)
d+ 1
∥∥∥∥∥ = 1: d
and
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M G FM NM PM
Ai
LMi
Gi
d 1d+1d−1
a
daR
(d+1)d(d−2)d−1
Rd
Rd b
b(d− 1)
Figure 3: Points on the Euler line and Feuerbach sphere, and ratios of line segments.
‖NM − FM‖ : ‖FM −M‖ =
∥∥∥M +∑d
i=0(Ai−M)
d−1 −M −∑d
i=0(Ai−M)
d
∥∥∥∥∥∥M +∑d
i=0(Ai−M)
d −M∥∥∥
=
∥∥∥∥∥∑d
i=0(Ai −M)
d(d− 1)
∥∥∥∥∥ :
∥∥∥∥∥∑d
i=0(Ai −M)
d
∥∥∥∥∥ = 1: (d− 1),
proving the remaining statements.
Remark 4.5. As noted in the Introduction, the sphere construction has been done forthe Euclidean case in several earlier works, giving a 3(d + 1)-sphere. In Minkowskispace, we ”lose” the (d + 1) points which are orthogonal projections of the LM
i ontothe facets Fi. In the planar case, this has already been pointed out in [8, 34].
Remark 4.6. Consider the (d + 1)-dimensional spatial representation of this config-uration where the segments between M and the vertices of our simplex are projectionsof some segments spanning a (d+ 1)-dimensional parallelepiped. Then, the Euler line〈MPM 〉 corresponds to the projection of the main diagonal of the parallelepiped, andthe points dividing the main diagonal in the ratio 1 : d, 1 : (d − 1), and 1 : (d − 2)project to the centroid, the center of the Feuerbach-2(d + 1)-sphere, and the Mongepoint, respectively.
Since it can be shown that NM divides the segment [FMM ] externally in the ratio 1 :d, i.e., [FMM ] is divided harmonically by G and NM , we obtain the following corollary,
12
the second statement of which has been noted in [9] for Euclidean orthocentric simplicesand the orthocenter. For a strictly convex normed plane (d = 2), the second statementcan be found in [34, Theorem 4.6].
Corollary 4.7. The Monge point NM associated to a circumcenter M of a d-simplexT is the center of homothety between the Feuerbach-2(d+1)-sphere centered at FM andthe circumsphere centered at M , with homothety ratio 1 : d. For any line from NM
meeting the associated circumsphere of T in Q, the point P dividing [NMQ] internallyin the ratio 1 : (d−1) is located on the Feuerbach sphere; conversely, for any line fromNM meeting the associated Feuerbach sphere in P , the point Q dividing the segment[NMP ] externally in the ratio d : (d− 1) is located on the circumsphere of T .
We conclude this Section with a Corollary which is an immediate consequence ofthe affine nature of Theorem 4.2.
Corollary 4.8. In a d-simplex in Minkowskian d-space, the points M , G, FM , NM ,PM are either collinear (on the Euler line), or they all coincide.
In the latter case, instead of speaking of the Euler line not being well-defined, some-times the term collapsing Euler line is used.
5 Generalizations for polygons in the plane
Generalizations of the concept of Euler line and Feuerbach circle have not just focusedon raising the dimension of the space; there have also been attempts to generalize topolygons. We will now see that easy generalizations arise if we consider such polygonsas projections of higher-dimensional simplices or sections of parallelepipeds. Thisrelates to descriptive geometry (see also Remark 4.6).
B. Herrera Gomez [24] and S.N. Collings [12] have written about remarkable circlesin connection with cyclic polygons in the Euclidean plane. Their definition of cyclicpolygon as a polygon possessing a circumcircle is directly extendable to any normedplane. Necessarily, cyclic polygons are convex.
Let P = conv{A0, . . . , Ad}, d ≥ 3, be a cyclic polygon with circumcenter M inthe normed plane (R2, ‖ · ‖). We may view the vertices of P as the images underprojection of certain vertices of a (d + 1)-dimensional parallelepiped Q in (d + 1)-dimensional space to an affine plane (which we then endow with the norm ‖ · ‖),namely the vertices adjacent to M ′ where M ′ projects to M (compare Remark 4.6).This makes P the projection of that hyperplane section P ′ of Q which is defined byall the vertices adjacent to M ′. Alternatively, we may view P as the shadow of ad-simplex T , which itself is a projection of the hyperplane section P ′ of Q to an affine(d− 1)-subspace.
We now define the points PM (complementary point), NM (Monge point), G (cen-troid), FM (2(d+ 1)-center) of the polygon to be the respective parallel projections ofthe following distinguished points on the main diagonal of the parallelepiped, whichwould have the corresponding meaning for the d-simplex T when M ′ projects to acircumcenter of T , see Section 4. That is,
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G =
∑di=0Ai
d+ 1= M +
∑di=0(Ai −M)
d+ 1is called the centroid of the polygon P,
FM = M +
∑di=0(Ai −M)
dis called the 2(d+ 1)-center of the polygon P,
NM = M +
∑di=0(Ai −M)
d− 1is called the Monge point of the polygon P,
PM = M +
d∑i=0
(Ai −M) is called the complementary point of the polygon P.
These points either coincide or are collinear on the Euler line of the polygon P(compare Corollary 4.8), with the division ratios given in Theorem 4.2. We can theneasily deduce the following relationships.
Theorem 5.1. Let P = conv{A0, . . . , Ad}, d ≥ 3 be a cyclic polygon with circumcen-ter M and circumradius R in the normed plane (R2, ‖ · ‖). Then:(a) The complementary point PM is common to all the circles S(P i
M , R), i = 0, . . . , d,where P i
M is the complementary point of the subpolygon Pi = conv ({A0, . . . , Ad} \ {Ai})with respect to the circumcenter M .(b) The lines 〈AiP
iM 〉 are concurrent in CM , where CM := M + 1
2
∑di=0(Ai −M) is
the midpoint of [MPM ] and called the spatial center of P with respect to M .(c) The midpoints Ei of the segments joining the vertices Ai, i = 0, . . . , d, with thecomplementary point PM are concyclic in the circle S(CM ,
R2 ).
(d) The point CM is common to all the circles S(CiM ,
R2 ), where Ci
M is the spatialcenter of the subpolygon Pi with respect to the circumcenter M , i = 0, . . . , d, and thepoints Ci
M also lie on the circle S(CM ,R2 ).
Proof. We have
PM = M +
d∑j=0
(Aj −M) = M +
d∑j=0j 6=i
(Aj −M) + (Ai −M) = P iM + (Ai −M).
Since (Ai − M) is a radius of any translate of the circle S(M,R), we obtain thestatement in (a). In the spatial representation in (d+1)-dimensional space, the vertexprojecting to the complementary point PM is the endpoint opposite M ′ of the maindiagonal of the parallelepiped Q (i.e., the line which projects to the Euler line), whereasthe pre-images of the points P i
M are vertices adjacent to the pre-image of PM . Thusthe pre-images of each point P i
M and Ai, i = 0, . . . , d, together span another maindiagonal of the parallelepiped Q. The main diagonals of the parallelepiped intersectin one point C ′ (the centroid of the parallelepiped), and this point halves each maindiagonal. The projection of this point is the point CM by definition, which provespart (b). Note that at most d − 1 of the lines 〈AiP
iM 〉 may not be well-defined, and
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precisely when their pre-images are parallel to the null space of the projection, but atleast 2 lines remain to determine the point CM . Part (d) is similar to part (a), in that
CM = M +1
2
d∑j=0
(Aj −M) = M +1
2
d∑j=0j 6=i
(Aj −M) +1
2(Ai −M) = Ci
M +1
2(Ai −M).
The second statement in (d) follows trivially. Finally, for part (c), consider Figure 3and observe that the line 〈CMEi〉 is parallel to 〈MAi〉 for each i = 0, . . . , d.
Remark 5.1. For the Euclidean plane and d = 3, part (d) is well known [50, pp.22–23]. For all d ≥ 3, the statements (a)–(c) have been established in [24] wherePM is called the orthocenter, and S(CM ,
R2 ) the Feuerbach circle of the polygon. For
strictly convex normed planes part (d) has been shown in [34, Theorem 4.18], callingthe point CM the center of the Feuerbach circle S(CM ,
R2 ), and the circles S(Ci
M ,R2 )
the Feuerbach circles of the subpolygons. The motivation in either case was to observe aradius half as long as the radius of the original circumcircle. We see that the statementsextend in some way to all Minkowski planes, though one has to be careful in theirformulation; recall that in planes which are not strictly convex, we cannot necessarilyspeak of the (unique) circumcircle, or the (unique) intersection of several circles.
Remark 5.2. Note that M is a circumcenter of P , and also a circumcenter for eachof its sub-polygons with d ≥ 3 vertices. The analogous statement for a d-simplex in d-space is wrong, i.e., a circumcenter of a d-simplex T is not a circumcenter for each ofits facets, which is the reason for the lack of analogous higher-dimensional statementsinvolving the complementary points of facets of T in Section 4.
An alternative, equally plausible definition of (orthocenter and) Feuerbach circle ofa polygon in the Euclidean plane was given by Collings [12]. This, too, generalizesto normed (Minkowski) planes, and is easily provable using the spatial representationgiven above. Both concepts of Feuerbach circles are illustrated in Figure 4, for cyclicpentagons in the `1-norm.
Theorem 5.2. Let P = conv{A0, . . . , Ad}, d ≥ 3, be a cyclic polygon with circumcen-ter M and circumradius R in the normed plane (R2, ‖ · ‖).(a) The Monge point NM is the point of intersection of the lines 〈AiN
iM 〉, i = 0, . . . , d,
where N iM is the Monge point of the subpolygon Pi = conv ({A0, . . . , Ad} \ {Ai}).
(b) The centroids Gi of the subpolygons Pi = conv ({A0, . . . , Ad} \ {Ai}), i = 0, . . . , d,are concyclic on S(FM ,
Rd ), where FM is the 2(d + 1)-center of the polygon. Further-
more, the circle S(FM ,Rd ) passes through the (d+ 1) points LM
i dividing the segments[NMAi] in the ratio 1: (d− 1).(c) The Monge points N i
M of the subpolygons are concyclic on the circle
S
M +1
d− 2
d∑j=0
(Aj −M),R
d− 2
15
with its center on the Euler line.
Proof. We have
NM = M +1
d− 1
d∑j=0
(Aj −M) = Ai +d− 2
d− 1
M +
1
d− 2
d∑j=0j 6=i
(Aj −M)
−Ai
= Ai +
d− 2
d− 1
(N i
M −Ai
),
which proves part (a). Part (b) is clear with Theorem 4.3 and the fact that thesegments [FMGi] and [FML
Mi ] have equal length and are homothets of [MAi] for each
i = 0, . . . , d (with factor 1d and homothety center NM ). For part (c), observe that
for each i = 0, . . . , d, N iM is the intersection of the lines 〈MGi〉 and 〈AiNM 〉, see also
Figure 3. Since the above equation shows that NM divides the segment [AiNiM ] in the
ratio (d − 2) : 1, the homothet of the circumsphere with respect to homothety centerNM and homothety ratio − 1
d−2 passes through the N iM . Thus the corresponding center
can be calculated as M + 1d−2
∑dj=0(Aj −M) (on the Euler line), and the radius is
Rd−2 .
Remark 5.3. Collings [12] proved a variant of part (a) for the Euclidean planeand called the point NM differently, namely the orthocenter of the polygon. In fact,Collings’ orthocenter (per our definition, the Monge point NM ) was defined induc-tively, using the base case d = 2, i.e., starting at sub-triangles of P , whose Mongepoint, complementary point, and C-orthocenter coincide. Note that an inductive defi-nition of the Monge point as such necessitates that M is the circumcenter at each stageof the recursion (otherwise the resulting points at each stage would not correspond toour definition of Monge point), and thus only works in the plane. In the context ofd-simplices, we did not consider this recursion for precisely this reason (although ofcourse, the respective lines exist in higher-dimensional space, and they are concurrentat the corresponding points!).
Remark 5.4. Part (b) was also proved for the Euclidean plane in [12], and in anal-ogy with the nine-point-circle of a triangle, the circle S(FM ,
Rd ) was named the (gen-
eralized) nine-point-circle, although it was only observed to pass through the (d + 1)centroids Gi. B. Herrera Gomez [24] extended the statements, for example by proving(c) for the Euclidean plane, and by investigating related infinite families of circles.
6 Concluding remarks and open problems
Solutions to questions from Elementary Geometry in normed spaces often yield aninteresting tool and form the first step for attacking problems in the spirit of Discrete
16
M
CM
PM
A0
P 0M
C0M
E0
(a) The Feuerbach circle of half size.
MFM
PM
NMG0
A0
LM0
(b) The Feuerbach circle of Collings.
Figure 4: Comparison of different definitions of the Feuerbach circle for a pentagon inthe `1-norm. For (a) see Remark 5.1, and for (b) see Remark 5.4. Respectiveradii are marked by dotted line segments, and the relevant part of the Eulerline [MPM ] is marked in bold. One pair of special points on the Feuerbachcircle is constructed in each case (with thin solid auxiliary lines).
17
and Computational Geometry in such spaces (see, e.g., [3, 4] for the concepts of cir-cumballs and minimal enclosing balls, or [31, Section 4] referring to bisectors as basisof an approach to Minkowskian Voronoi diagrams). And of course it is an interestingtask for geometers to generalize notions like orthogonality (see [1,2,5]), orthocentricity(cf. [8, 34, 40, 45]), isometries (see [36, 38]), and regularity (see [37]) in absence of aninner product. In case of regularity, we may ask which figures are special, and whatare useful concepts to describe their degree of symmetry in normed planes and spaces?For Minkowski spaces nothing really satisfactory is done in this direction, and it isclear that a corresponding hierarchical classification of types of simplices would yieldthe first step here. Thus, it would be an interesting research program to extend thegeneralizable parts of the concepts investigated in [14–16] to normed spaces: whatparticular types of simplices are obtained if special points of them, called ”centers”(like circum- and incenters, centroids, Monge points, Fermat-Torricelli points etc.),coincide or lie, in cases where this is not typical (e.g., in case of the incenter), on theEuler line? In view of [36, 38], a related interesting task might be the development ofsymmetry concepts based on Minkowskian isometries.
Another interesting point of view comes in with the field of geometric configu-rations which is summarized by the recent monograph [21]. Namely, the Three-Circles-Theorem and Miquel’s Theorem can be successfully extended to normed planes(see [8,34,47] and thus have acquired some recent popularity. Clifford’s circle configu-ration, for circles of equal radii also called Clifford’s Chain of Theorems (see [33,51]), isa direct generalization of the Three-Circles-Theorem and also part of the collection oftheorems which nicely ties to visualizations of the Euler line and the Feuerbach circlein the spirit of descriptive geometry (see our discussion at the beginning of Section 5above). Based on [8,34], Martini and Spirova extend in [35] the Clifford configurationfor circles of equal radii to strictly convex normed planes, and prove properties of theconfiguration as well as characterizations of the Euclidean plane among Minkowskiplanes. Using our terminology from Section 5 above, one may easily color the verticesof the parallelepiped Q alternatingly red and blue, with M ′ being blue. Then theprojected blue vertices are centers of circles of the Clifford configuration, whereas theprojected red vertices are in the intersection of certain subsets of the circles. Due tothe successful extension of these topics to normed planes and spaces one might hopethat also further configuration concepts can be generalized this way. E.g., one cancheck whether the comprehensive geometry of n-lines (which are the natural exten-sions of complete quadrilaterals; see Section 4 of the survey [32]) and systems of circlescorresponding with them contain parts which are generalizable this way.
As basic notions like isoperimetrix (see [49, §4.4 and §5.4]) demonstrate, duality(of norms) plays an essential role in the geometry of normed spaces. This conceptshould also be used in that part of Minkowski Geometry discussed here. It should bechecked how far this important concept can be used to get, in correspondence withalready obtained results, also “dual results”, such that for example results on notionslike “circumball” and “inball” might be dual to each other.
Finally we mention that still for the Euclidean plane there are new generalizationsof notions, such as generalized Euler lines in view of so-called circumcenters of massetc. (see [48]), which could, a fortiori, also be studied for normed planes and spaces.
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Undine LeopoldTechnische Universitat ChemnitzFakultat fur MathematikD - 09107 ChemnitzGermany
Horst MartiniTechnische Universitat ChemnitzFakultat fur MathematikD - 09107 ChemnitzGermany
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